On the well-posedness of Bayesian inversion for PDEs with ill-posed forward problems

We study the well-posedness of Bayesian inverse problems for PDEs, for which the underlying forward problem may be ill-posed. Such PDEs, which include the fundamental equations of fluid dynamics, are characterized by the lack of rigorous global existence and stability results as well as possible non-convergence of numerical approximations. Under very general hypotheses on approximations to these PDEs, we prove that the posterior measure, expressing the solution of the Bayesian inverse problem, exists and is stable with respect to perturbations of the (noisy) measurements. Moreover, analogous well-posedness results are obtained for the data assimilation (filtering) problem in the time-dependent setting. Finally, we apply this abstract framework to the incompressible Euler and Navier-Stokes equations and to hyperbolic systems of conservation laws and demonstrate well-posedness results for the Bayesian inverse and filtering problems, even when the underlying forward problem may be ill-posed.